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Answers to Odd Problems
Answers are not given when they reveal to much about the solution. Some decimals have been truncated rather than rounded
1. Basic Concepts
1, Ω = {ABC, ACB, BAC, BCA, CAB, CBA}. 3, 1/4. 5, (a) 16/36, (b) 20/36. 7, 3/4. 9, 1:4/57, 2:16/57, 3:15/57, 4:12/57. 11, Number of outcomes out of 216: (a) 1, (b) 3, (c) 6, (d) 10, (e) 15, (f) 21, (g) 25, (h) 27. 13, 90. 15, 77. 17, 0.2. 19, Yes. 21, Yes. 25, 0.44. 27, 19/27. 29, (a) 0.048, (b) 0.296, (c) 0.464, (d) 0.192. 31, 0.4032. 33, 0.4914. 35, 0:6/36, 1:10/36, 2:8/36, 3:6/36, 4:4/36, 5:2/36. 37, The same as for the sum of two ordinary dice. 39, 7. 41, 1/4. 43, − 17 /216. 45, Flag or Joker: − 13 /54, 20: − 12 /54, 10: − 10 /54, 5: − 12 /54, 2: − 9 /54, 1: − 6 /54. 47, 32/10. 49, 26, 51, mean 5.8125, variance 1.03. 53, Mean 3.888, variance 0.432. 55, No.
2. Combinatorial Probability
1, 9!. 3, 3360. 5, (a) 151,200, (b) 210. 7, 55. 9, (a) 1. 7576 × 107 , (b) 0.6391. 11, 0.0605. 13, (a) 540, (b) 372. 15, 118,800. 17, (a) 240, (b) 480. 21, (a) 120, (b) 60, (c) 504,000, (d) 34,650. 25, 103,680. 27, (a) 24, (b) 576, (c) 105, (d) 2,520, (e) 40,320, (f) 384. 29, 60/1024. 31, (a) 0.152, (b) 0.618. 33, (a) 0.05469, (b) 0.92981. 35, 21/32. 37, (a) 0.3874, (b) 0.3773, (c) 0.3725. 39, 0.3297. 41, Poisson: 0.778801, Exact: 0.778703. 43, 0.1889. 45, 0.9735. 47, 0.4232. 49, 0.1119. 51, 0.6664. 53, 10/21. 55, 0.3343. 57, 0.4431. 59, 0.3685. 61, (a) 0.4085, (b) 0.5107, (c) 0.0766. 63, 23,950,080. 65, 5. 4703 × 10 −^4. 67, 1/2. 69, 0.5814. 71, 0.2429. 73, (a) 0.7969, (b) 0.1992, (c) 0.0038. 75, Number of outcomes out of 7776: (a) 6, (b) 150, (c) 300, (d) 1200, (e) 1800, (f) 3600, (g) 720. 77, 6/32. 79, (a) 0.9, (b) 0.3. 81, 0.3439. 83, 0.028. 85, P (A) = 0.1655, P (A)/P (B) = 3.9971. 87, first bound: 0.1667, second: 0.1551, third: 0.1555213, exact: 0.1555124.
3. Conditional Probability
1, 2/3. 3, 4/25. 5, (a) 0.4385, (b) 0.4561, (c) 0.1052. 7, 6/7. 9, 2/3. 11, 1/3. 13, P (A ∩ B) = 1/8, P (B) = 3/8, P (A ∪ B) = 1/2. 15, 0.6. 19, 11/216. 21, 14: 146/1296. 23, 0.9568. 25, 0.17. 27, 2/5. 29, 0.7. 31, 0.4918. 35, 1/51. 37, (a) 0.9, (b) 1/730. 39, 12/29. 41, 8/9. 43, 56/65. 45, 4/5. 47, 1/7. 49, (a) 1/2, (b) 5/9. 51, 8/9. 53, 5/6. 55, 4/7. 57, 4/7, 2/7, 1/7. 59, 0.062, 0.3387. 63. (a) 3/
(b) 1/2,
Y X=1 2 3
4. Markov chains
1, (a) x = 0.4, y = 0.4, z = 0.6, (b) x = 0.2, y = 0.5, z = 0.2. 3, 0.294. 5, 13/32. 7, (b) 0.22, 0.166. 9, 51.2% in 1980, 56.8% in 1990, and 60.9% in 2000. 11, (a) 0.55, (b) 0.575, (c) 0.6. 13, 0.7825, 0.727291, 8/11 = .0727272. 15, 38%, 25%, 8/33 = 24%. 17, 0.211, 0.286, 0.502. 19, 4/19. 21, (b) 7/13. 23, 4/5. 25, 0.25, 0.5, 0.25. 27, (a) 0.35, 0.34, 0.31, (b) 0.5294, 0.3235, 0.1470. 29, Long run frequencies are A:0.2, C:0.55, T:0.25. 31, (b) 0.2321, 0.1964, 0.5714. 33, 0.2817. 35, π(0) = 100/122, π(1) = 10/122, π(2) = 2/122, π(3) = 10/122. 37, π(0) = 0.1, π(1) = 0.4, π(2) = 0.3, π(3) = 0.2. 39, 1/3. 41, 16.666. 43, (a) 1/4, (b) 3.916. 45, 0.125/0.685.
5. Continuous distributions
1, c = 1/9. 3, (a) 1/2, (b) 0.3, (c) 0.05. 5,(a) 15/28, (b) 127/49, (c) 2.304. 7, Yes, x−^2 e−^1 /x. 9, (a) x^2 /4 for 0 ≤ x ≤ 2, (b) 1/4, (c) 7/16, (d)
(a) x^1 /^2 for 0 ≤ x ≤ 1, (b) 1 −
3 /2, (c) 1/6, (d) 1/4. 13, any number in [0, 1]. 15, (a) 1 − x−^3 , (b) (1 − u)−^1 /^3. 21, y−^1 /^2 /2. 23, 1 − exp(−e−x). 25, (a) f (y) + f (−y) for 0 ≤ y ≤ 1, (b) {f (
z) + f (−
z)}/ 2
z for 0 ≤ z ≤ 1. 27, (a) c = 1, (b) 3/8. 29, (1 − z)^2 /2 for 0 ≤ z ≤ 1. 31, 7/16. 33, F (x, y) = min{x, y} if x, y > 0 and min{x, y} ≤ 1, F (x, y) = 1 if min{x, y} > 1, 0 otherwise. 35, FX (x) = FX,Y (x, ∞) = limy→∞ FX,Y (x, y). 37, (a) fX (x) = (2/π)
1 − x^2 , (b) fY (y|X = x) = 1/ 2
1 − x^2 for −
1 − x^2 < y <
1 − x^2. 39, fX (x) = 3(1−x)^2 for 0 < x < 1, fY (y) = 6y(1 − y) for 0 < y < 1, (b) fX (x|Y = y) = 1/(1 − y), 0 < x < 1 − y.
6. Limit theorems
1, −1. 3, − 3 /5. 5, 5.03. 7, 3.114. 9, 123.136. 11, Mean 6, variance 43. 13, (a) mean 2, variance 1.82, (b) 1 boy and 1 girl. 15, mean 14.7, variance 38.99. 17, (a) P (X ≥ 75) ≤ 2 /3, (b) P (40 < X < 60) ≥ .75. 19. Chebyshev: ≤ 1 /4, normal approx: 0.0455. 21, 0.484. 23, 0.041. 25, 0.0016. 27, 24. 29, 0.0085. 31. 0.1056. 33, (a) 0.2658, (b) 0.2565, (c) 0.2475. 35, (a) .0566, (b) .0884. 37, 157. 39, 373. 41, 0.3897. 43, 527. 45, 0.0116. 47, [0. 508 , 0 .528]. 49, [0. 0044 , 0 .0356]. 51, 52.33. 53, 0.0081. 55, Yes. 57, No. 59, 23.
